Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C22×C3×#477 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C3 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ C22×C3×C42 ◁ G | |
2358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2)(21,22,23);(3,4);(5,6,8,10)(7,11,12,9)(14,17)(15,18)(19,20);(5,7,8,12)(6,9,10,11)(13,14,16,17)(15,18,19,20);(13,15,16,19)(14,18,17,20)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53465);